Michael P. Clements and David I. Harvey 171There are a number of explanations as to why forecast combination works.
Perhaps the most common is the portfolio diversification argument that under-
pinned the original analysis of Bates and Granger (1969), as recently discussed by
Granger and Jeon (2004) and Timmermann (2006), amongst others. The idea is
simply that the individual forecasts are each based on partial, and incompletely
overlapping, information sets, as might be the case if they reflect private informa-
tion, for example. The degree of overlap in the information sets is key, as is apparent
in the discussion of Bates and Granger (1969) in section 4.2. An explanation
stressed by Hendry and Clements (2004) is that of forecasts based on misspeci-
fied models when there are structural breaks and, as noted by Timmermann (2006,
p. 138), a number of other papers discuss the roles of model misspecification and
structural breaks. Hendry and Clements (2004) and Timmermann (2006) discuss a
number of other reasons that would justify pooling.
Our survey is the latest of a number of recent reviews of the large literatures on
the topics of forecast combination and on forecast encompassing. These include
Clemen (1989), Diebold and Lopez (1996), Newbold and Harvey (2002) and
Timmermann (2006). One of the key ways in which it differs from the others
is the emphasis on the testing of forecast encompassing alongside the treatment of
forecast combination. We are also able to include some of the important develop-
ments that have only recently found their way into the literature. For expositional
convenience we focus on two forecasts, but in general more than two forecasts
may be combined, and the notion of forecast encompassing can be generalized to
the case of multiple forecasts (see Harvey and Newbold, 2000).
The plan of the rest of the chapter is as follows. Section 4.2 outlines the historical
development of forecast combination and encompassing, and fills in some of the
details. Section 4.3 describes the key developments when the forecasts are based on
models and one wishes to compare the forecasting models. As the forecasts are gen-
erated from models in which the unknown parameters are replaced with estimates,
an allowance is made for the true values having been replaced by random variables
when the forecasts are compared. Section 4.4 considers forecasting from nested
models, which is not covered by the analysis in section 4.3 and requires a separate
treatment. Section 4.5 describes forecast encompassing tests within a framework of
conditional testing of predictive ability, and where the emphasis shifts to testing
forecastingmethodsrather thanmodels. Thus far, we have maintained an assump-
tion of symmetry of the loss function: the implications of dispensing with this
assumption form the material of section 4.6. Section 4.7 offers some concluding
remarks.
4.2 Historical development
4.2.1 Forecast combination
The notion of combining different forecasts of the same quantity in order to
improve predictive accuracy was first proposed by Bates and Granger (1969).
Suppose we have available twoh-steps-ahead forecasts,f 1 tandf 2 t, of the quantity
yt. In this section, in line with the early literature on forecast combination (and